Simplify the following expression and state the condition under which the simplification is valid. $a = \dfrac{3q^2 - 48q + 192}{-5q^2 + 70q - 240}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ a = \dfrac {3(q^2 - 16q + 64)} {-5(q^2 - 14q + 48)} $ $ a = -\dfrac{3}{5} \cdot \dfrac{q^2 - 16q + 64}{q^2 - 14q + 48} $ Next factor the numerator and denominator. $ a = - \dfrac{3}{5} \cdot \dfrac{(q - 8)(q - 8)}{(q - 8)(q - 6)}$ Assuming $q \neq 8$ , we can cancel the $q - 8$ $ a = - \dfrac{3}{5} \cdot \dfrac{q - 8}{q - 6}$ Therefore: $ a = \dfrac{ -3(q - 8)}{ 5(q - 6)}$, $q \neq 8$